Permutability of subgroups of \(G\times H\) that are direct products of subgroups of the direct factors (Q5956891)

Permutability within a direct product of finite groups is considered. The author is interested in characterising permutable subgroups of \(G\times H\) which are direct products of the direct factors, that is, subgroups that are equal to \(A\times B\), with \(A\leq G\) and \(B\leq H\). Under these hypotheses, a necessary condition for \(A\times B\) to be a permutable subgroup of \(G\times H\) is that \(A\) and \(B\) are permutable subgroups of \(G\) and \(H\), respectively. However, some examples, as the direct product of the modular non-Abelian group of order \(27\) and the cyclic group of order \(25\), show that this condition is not sufficient. It is proved that \(A\times B\) is permutable in \(G\times H\) if and only if \(A\) is a permutable subgroup of \(G\times(H/\text{Core}_H(B))\) and \(B\) is a permutable subgroup of \((G/\text{Core}_G(A))\times H\). With this conclusion, the author studies when a subgroup \(M\) of \(G\) is permutable in \(G\times H\). Again, a necessary condition is that \(M\) must be permutable in \(G\), but it is not sufficient: as shown in Example~7.1, there may be permutable subgroups of \(G\) which are not permutable in \(G\times H\). The following result is proved: If \(M\) is a permutable subgroup of \(G\) and \(H\) is a group, then \(M\) is a permutable subgroup of \(G\times H\) if and only if for each prime \(p\) dividing \(|H|\) and \(g\in G\) of \(p\)-power order, \(g\in N_G(\langle g^{\exp P}\rangle M)\), where \(P\) is any Sylow \(p\)-subgroup of \(H\).